Written by Muchang B. (KIS'19)
━━ August 18th, 2018 ━━
Introduction to the Fourier Transform
The universe is filled with waves every second. From electromagnetic waves in the form of light to the movements of water in a pool, we humans experience them every minute of our lives. More specifically, the variety of sounds that we hear every day can be observed as a mixture of simple harmonic sound waves, all overlapping together through constructive and destructive interference, analogous to how differently colored paints are mixed in a bucket. Even with 4 different harmonic waves of the same amplitude, one can create a complex wave that is hard to define by itself without the given “ingredient” waves. This “resultant wave” would be even more complex if the ingredient waves had different amplitudes (or levels of loudness).
With the “mixed” waves that we hear every day, the fourier transform serves as a system to derive the ingredient waves from the resultant wave, or to unmix the different colored paints in the bucket. This paper will serve to explain the properties and applications of the fourier transform mainly through both a visual and a mathematical approach.
Before we get into the general approach, let us start with a simple harmonic wave with a frequency of 3 beats per second, indicated by y=cos(6πx)+1.
If we were to wrap a finite portion of this graph around a polar function such that the y-value was equivalent to the r-value and the x-value was a scaled version of the θ-radian-value (scaled by h), we would be able to stretch this new graph around such that its frequency around the polar graph would be k cycles per second. The polar equation for this graph would be r=cos(6πhθ)+1. In this case, I’ve used the graph from -1≤ x ≤10 and scaled it so that h=0.1, 0.3, 0.7, and 2, consecutively from left to right.
Thus, this graph would have two frequencies: one determined by the rectangular form of the graph itself, and one determined by the stretch factor of the graph around the polar origin. The graph looks chaotic at most points for y=cos(6πhx)+1 except at some values of h (at h=0.0531, 0.1061, 0.1592, 0.2122 from left to right), where the graph seems to nicely overlap into a neat array. From left to right, the polar frequency of the graphs are 3 cycles/sec, 1.5 cycles/sec, 1 cycle/sec, and 0.75 cycles/sec.
We also realize that the graph with 3 cycles per second is in this collection of graphs, and additionally, the one quality that distinguishes these graphs is that it is a limacon (more specifically, a cardioid). Therefore, by looking at the polar frequency of the cardioid function, we can see what the rectangular frequency of the simple harmonic graph is.
However, for future reasons, let us use another characteristic to differentiate the graph with 3 cycles per second. Instead, let us look at the “center point” of the graph, indicated by its center of mass. Again, we see that the center of mass of the cardioid is quite peculiar, since it is abnormally far away from the origin, unlike the others.
For simplicity, we can measure this distance by only focusing on the x-value of the center point (we will call this Xcenter). We can thus create another graph where the x-value is the polar frequency of the graph and the y-value is the Xcenter. This is the graph of the fourier transform. Except for when the x-value/polar frequency is near 0 (when the graph is all bunched up near θ=0), the only other “spike” in the fourier transform is at x=3. From this result, we may be able to hypothesize that the maximum values in the peaks of this fourier transform graph reveals the frequencies of the ingredient waves.
At this point, it is helpful to think of the fourier transform as a function of a wave graph, similar to taking the derivative or the integral of a function. If the fourier transform is a function p(x), then it inputs a function f(x) and outputs the result p(f(x)).
Since the fourier transform is an operation (like taking the derivative of a graph to get another graph), adding the two original graphs and taking the fourier transform of the sum is the same as taking the sum of the two individual fourier transforms.
With both a mathematical and visual understanding of the Fourier transform, we can now transition to connecting this to the Heisenberg Uncertainty principle, which states that for a particle, you cannot measure its exact momentum and velocity at the same time. In other words, as you measure the position of a particle more and more accurately, your measurement of its momentum (or velocity) becomes less and less accurate. We must state 3 pieces of information before we can move on to connecting these two concepts.
First is the dual nature of all matter, also known as the wave-particle duality, which states that every particle/quantic entity may be described in waves. Second is Fourier’s Theorem (a statement proven by Jean Baptiste Fourier) which states that all periodic waves of a certain frequency can be described as the summation (called the Fourier Series) of simple harmonic waves. Finally, Broglie’s equation on the inverse relationship between the wavelength and momentum of a wave must be accepted. This equation is: p=h/λ, where h is Planck's constant, p is momentum, and λ is wavelength.
Let us first focus on the location of a particle, which can be described with waves using superposition, which essentially describes the probability of a particle being located in space, according to quantum mechanics. In the diagram below, this describes the superposition of a particle in a 1-D line. The points in space with the greatest amplitude is known to be the points where the particle has the highest probability of being located, while the points with low amplitudes have the least likelihood of the particle being located on. The first scenario shows a particle with an almost definite location, as indicated by the wave’s huge spike in the middle indicating its almost certain superposition in that point.
This next scenario shows there being absolutely no certainty in where the particle is located due to it being a simple harmonic wave without any spikes indicating likelihood of its superposition.
However, this second case shows a definite wavelength of the wave, which is at 2π, implying that according to Broglie’s equation, its momentum (and its velocity) can be accurately determined. Thus, a higher accuracy in velocity is determined, with no accuracy in the particle’s superposition. On the contrary, the first wave shows a definite superposition of the particle, but due to the lack of a constant wavelength, it gets harder to define it and thus an approximation must be made. Therefore, the wave-particle duality of matter implies that accurately measuring both a particle’s superposition and its momentum at once is impossible (and is more of a trade-off).
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Credits:
Picture Credits go to Grant Sanderson, or 3Blue1Brown (for the 3 pictures with black backgrounds)
Some of the content is also credited to Grand Sanderson, or 3Blue1Brown for his visual explanation of the fourier transform in his video: https://www.youtube.com/watch?v=spUNpyF58BY
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